You are right, Goyalsau, to be thinking about the relationship between
the lengths of the sides, but because the triangles are parts of a
rectangle, they are right triangles so we can use a more specific rule
than the sum of two sides being greater than the third side. We can
use the Pythagorean theorem, as Euro does, to relate the sides
directly.
From the diagram, we can see a right triangle whose hypotenuse is the
diagonal of the rectangle; let's call that d. The legs of the triangle
are of length 5 and x, so the Pythagorean theorem states 5^2 + x^2 =
d^2. That means d = sqrt(5^2 + x^2). Next, recall that 5, 12, 13 is a
Pythagorean triple-this can help us avoid some nasty computations.
Since 5-12-13 is a Pythagorean triple, in order for d to EQUAL 13, x
will have to be equal to 12. So for d to be less than 13, x will have
to be less than 12.
Statement 1 says that x < 12, so 5^2 + x^2 < 5^2 + 12^2 = 13^2, so the
diagonal must be less than 13. Statement 1 is sufficient.
Statement 2 only says the diagonal of the rectangle is greater than
10. It could be 10.5 or 11, which are less than 13, but it could also
be 14 or 15. Since we can't know whether the diagonal is less than 13,
the statement is insufficient.
Choice A is correct.
